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Exercise Hours Introductory statistics students fill out a survey on the first day of class. One of the questions asked is "How many hours of exercise do you typically get each week?" Responses for a sample of 50 students are introduced in Example 3.25 on page 207 and stored in the file ExerciseHours. The summary statistics are shown in the computer output. The mean hours of exercise for the combined sample of 50 students is 10.6 hours per week and the standard deviation is 8.04 . We are interested in whether these sample data provide evidence that the mean number of hours of exercise per week is different between male and female statistics students. Variable Gender N Mean StDev Minimum Maximum \(\begin{array}{lllllll}\text { Exercise } & \text { F } 30 & 9.40 & 7.41 & 0.00 & 34.00\end{array}\) \(\begin{array}{llll}20 & 12.40 & 8.80 & 2,00\end{array}\) Discuss whether or not the methods described below would be appropriate ways to generate randomization samples that are consistent with \(H_{0}: \mu_{F}=\mu_{M}\) vs \(H_{a}: \mu_{F} \neq \mu_{M} .\) Explain your reasoning in each case. (a) Randomly label 30 of the actual exercise values with " \(\mathrm{F}^{\prime \prime}\) for the female group and the remaining 20 exercise values with " \(\mathrm{M} "\) for the males. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\) (b) Add 1.2 to every female exercise value to give a new mean of 10.6 and subtract 1.8 from each male exercise value to move their mean to 10.6 (and match the females). Sample 30 values (with replacement) from the shifted female values and 20 values (with replacement) from the shifted male values. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\) - (c) Combine all 50 sample values into one set of data having a mean amount of 10.6 hours. Select 30 values (with replacement) to represent a sample of female exercise hours and 20 values (also with replacement) for a sample of male exercise values. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\)

Step by step solution

01

Evaluation of Method (a)

Method (a) suggests to simply relabel 30 of the actual exercise values for the female group and the remaining 20 exercise values for the males, then compute the difference in the sample means. This method looks at the difference between means but it neglects the actual gender-based data distribution. The sample sizes for males and females are different and the exercise hours for each gender might have a different distribution. Therefore this method is not appropriate for evaluating the null hypothesis.

02

Evaluation of Method (b)

Method (b) suggests adjusting every female exercise value by adding 1.2 to give a new mean of 10.6 and subtract 1.8 from each male exercise value to adjust their mean to 10.6 to match the females. Then sample 30 values (with replacement) from the adjusted female values and 20 values (with replacement) from the adjusted male values and compute the difference in the sample means. This method, while attempting to establish a common mean and evaluating the mean differences, disrupts the original data distribution which can introduce biases. Therefore, it is also not appropriate.

03

Evaluation of Method (c)

Method (c) suggests combining all 50 sample values into one set of data having a mean amount of 10.6 hours. Then select 30 values (with replacement) for a sample of female exercise hours and 20 values (with the same replacement condition) for a sample of male exercise values and compute the difference in the sample means. This approach respects the data distribution and the replacement condition provides better randomness, making it the most appropriate option to evaluate the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis, often denoted as \(H_0\), is a fundamental assumption in statistical hypothesis testing that proposes no difference or effect between certain characteristics of a population. In the context of exercise and gender, the null hypothesis would assert that male and female statistics students have the same mean number of exercise hours per week, symbolically written as \(\mu_{F} = \mu_{M}\). To test this assumption, we would compare the sample means from each gender group to see if any observed difference is statistically significant or if it could have occurred by random chance.

In evaluating different methods to test this hypothesis, care must be taken to ensure that the sample data aligns with the null hypothesis and that the method does not introduce any potential bias. For example, simply relabeling data points, without considering the original group distributions, might not respect the intent of \(H_0\) and could lead to incorrect conclusions.

Alternative Hypothesis

The alternative hypothesis \(H_a\) is used in statistical testing as a contrast to the null hypothesis. It represents the opposite of the null hypothesis and indicates that there is a statistically significant difference between the groups being compared.

In our exercise study example, the alternative hypothesis posits that there is a difference in the mean exercise hours per week between male and female statistics students, notated as \(\mu_{F} eq \mu_{M}\). If, after conducting an appropriate randomization test, we find that the evidence does not support the null hypothesis, we may then consider the alternative hypothesis to be more plausible. This would suggest pursuing further investigation into the factors leading to any observed discrepancy in exercise hours between genders.

Randomization Test

A randomization test, also known as a permutation test, is a non-parametric method to evaluate hypotheses. This type of test involves randomly rearranging the observed data and calculating the statistic of interest, like the difference in means, to create a distribution of possible outcomes under the null hypothesis.

In our textbook problem, method (c) provides an example of an appropriate randomization test. By combining all the exercise hours into one data set, and then resampling groups to represent males and females, we maintain the original distribution of data while testing for any significant differences between groups. This keeps the test aligned with \(H_0\) and allows us to see if the observed difference in sample means could have happened by chance, or if it might be evidence to support the alternative hypothesis, \(H_a\).

Sample Mean

The sample mean is a measure of central tendency that is calculated as the sum of all the observed values in a sample divided by the number of observations. It serves as an estimate for the population mean but can be subject to sampling variability.

For instance, in our statistical task, the sample mean is pivotal for understanding whether the average number of exercise hours per week may differ for male and female students. The computed sample mean reflects the central tendency of each subgroup and is used as a crucial component in hypothesis testing. It's also important to ensure that any methods used to generate randomization samples should maintain the integrity of these sample means. This emphasizes why method (b) was not appropriate, as it altered the original sample means artificially.

Standard Deviation

Standard deviation is a key statistic that measures the amount of variability or dispersion around the sample mean. A low standard deviation indicates that the data points are closer to the mean, whereas a higher standard deviation suggests a wider spread of values.

In the study comparing exercise hours, we look at the standard deviation to understand the spread of exercise hours around the mean for both male and female samples. When examining randomization methods, it's essential that the variability within the original data isn't compromised, as it could misrepresent the natural fluctuation of exercise hours. Given the important role of standard deviation in assessing data spread, any adjustments that could distort this measure (such as those suggested in method (b)) are generally considered inappropriate for hypothesis testing.